scholarly journals THE PROPORTIONAL COLORING PROBLEM: OPTIMIZING BUFFERS IN RADIO MESH NETWORKS

2012 ◽  
Vol 04 (03) ◽  
pp. 1250028 ◽  
Author(s):  
FLORIAN HUC ◽  
CLÁUDIA LINHARES SALES ◽  
HERVÉ RIVANO
Keyword(s):  
Mesh Networks ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Minimum Cost ◽  
Upper Bounds ◽  
Weighted Graphs ◽  
Chromatic Index ◽  
Lower And Upper Bounds ◽  
Coloring Problem ◽  
Wireless Mesh

In this paper, we consider a new edge coloring problem to model call scheduling optimization issues in wireless mesh networks: the proportional coloring. It consists in finding a minimum cost edge coloring of a graph which preserves the proportion given by the weights associated to each of its edges. We show that deciding if a weighted graph admits a proportional coloring is pseudo-polynomial while determining its proportional chromatic index is NP-hard. We then give lower and upper bounds for this parameter that can be computed in pseudo-polynomial time. We finally identify a class of graphs and a class of weighted graphs for which the proportional chromatic index can be exactly determined.

scholarly journals ESTRADA INDEX OF GENERAL WEIGHTED GRAPHS

2012 ◽  
Vol 88 (1) ◽  
pp. 106-112 ◽  
Author(s):  
YILUN SHANG
Keyword(s):  
Weighted Graph ◽  
Upper Bounds ◽  
Weighted Graphs ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Spectral Moments ◽  
Estrada Index ◽  
Low Order

AbstractLet $G$ be a general weighted graph (with possible self-loops) on $n$ vertices and $\lambda _1,\lambda _2,\ldots ,\lambda _n$ be its eigenvalues. The Estrada index of $G$ is a graph invariant defined as $EE=\sum _{i=1}^ne^{\lambda _i}$. We present a generic expression for $EE$ based on weights of short closed walks in $G$. We establish lower and upper bounds for $EE$in terms of low-order spectral moments involving the weights of closed walks. A concrete example of calculation is provided.

WEIGHTED-EDGE-COLORING OF k-DEGENERATE GRAPHS AND BIN-PACKING

2011 ◽  
Vol 12 (01n02) ◽  
pp. 109-124
Author(s):  
FLORIAN HUC
Keyword(s):  
Bin Packing ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Packing Problem ◽  
Bipartite Graphs ◽  
Specific Class ◽  
Weighted Graphs ◽  
Outerplanar Graphs ◽  
Underlying Graph ◽  
Multiple Edges

The weighted-edge-coloring problem of an edge-weighted graph whose weights are between 0 and 1, consists in finding a coloring using as few colors as possible and satisfying the following constraints: the sum of weights of edges with the same color and incident to the same vertex must be at most 1. In 1991, Chung and Ross conjectured that if G is bipartite, then [Formula: see text] colors are always sufficient to weighted-edge-color (G,w), where [Formula: see text] is the maximum of the sums of the weights of the edges incident to a vertex. We prove this is true for edge-weighted graphs with multiple edges whose underlying graph is a tree. We further generalise this conjecture to non-bipartite graphs and prove the generalised conjecture for simple edge-weighted outerplanar graphs. Finally, we introduce a list version of this coloring together with the list-bin-packing problem, which allows us to obtain new results concerning the original coloring for a specific class of graphs, namely the k-weight-degenerate weighted graph.

Complexity classification of the edge coloring problem for a family of graph classes

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Dmitriy S. Malyshev
Keyword(s):  
Edge Coloring ◽  
Complete Classification ◽  
Chromatic Index ◽  
Forbidden Subgraphs ◽  
Coloring Problem ◽  
Graph Classes ◽  
Complexity Classification ◽  
Index Problem

AbstractA class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain a complete classification of the complexity of the chromatic index problem for all monotone classes defined in terms of forbidden subgraphs having at most 6 edges or at most 7 vertices.

A NEW APPROACH FOR SOLVING THE MINIMUM COST FLOW PROBLEM WITH INTERVAL AND FUZZY DATA

2011 ◽  
Vol 19 (01) ◽  
pp. 71-88 ◽  
Author(s):  
MEHDI GHIYASVAND
Keyword(s):  
Flow Problem ◽  
Minimum Cost ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Minimum Cost Flow ◽  
Running Time ◽  
Minimum Cost Flow Problem ◽  
Cost Flow ◽  
Cost Scaling ◽  
The Cost

In particular, imprecise observations or possible perturbations mean that data in a network flows may well be better represented by intervals or fuzzy numbers than crisp quantities. In this paper we first consider the minimum cost flow problem with compact interval-valued lower and upper bounds, flows, and costs. We present a new method that shows this problem is solved using two minimum cost flow problems with crisp data. Then this result is extended to networks with fuzzy lower and upper bounds, flows, and costs. One of the best algorithms to solve the minimum cost flow problem with crisp data is the cost scaling algorithm of Goldberg and Tarjan.17 In this paper, the cost scaling algorithm is modified for fuzzy lower and upper bounds, flows and costs. The running time of the modified algorithm is equal to the running time of the cost scaling algorithm with crisp data.

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